Let $\phi$ be a compactly supported smooth function on $\mathbb{R}$. I'm looking for a simple proof that the operator
$$\left(-\frac{d^2}{dx^2}+x^2\right)^{-1}\phi$$ (where $x$ denotes multiplication by $x$) is a compact operator $H^1(\mathbb{R})\rightarrow L^2(\mathbb{R})$.
Added after: Is it true that $\left(-\frac{d^2}{dx^2}+x^2\right)^{-1}$ is a bounded operator $L^2(\mathbb{R})\rightarrow H^1(\mathbb{R})$? How to prove this?
Remark: I believe one can show this by some clever means, like using a special basis for $L^2(\mathbb{R})$ defined using Hermite polynomials, but I'm looking for something more down-to-earth, coming straight out of functional analysis.